Gorenstein Prüfer整环的一些新刻画

设R是整环.众所周知,R是Prüfer整环当且仅当每个可除模是FP-内射模当且仅当每个h-可除模是FP-内射模.本文引进了一种新的Gorenstein FP-内射模,并且证明了R是Gorenstein Prüfer整环当且仅当每个可除模是Gorenstein FP-内射模,当且仅当每个h-可除模是Gorenstein FP-内射模.     

Rings whose cyclic modules are lifting and ⊕-supplemented

It is proved that a semiperfect module is lifting if and only if it has a projective cover preserving direct summands. Three corollaries are obtained: (1) every cyclic module over a ring R is lifting if and only if every cyclic R-module has a projective cover preserving direct summands; (2) a ring R is artinian serial with Jacobson radical square-zero if and only if every (2-generated) R-module has a projective cover preserving direct summands; (3) a ring R is a right (semi-)perfect ring if and only if (cyclic) lifting R-module has a projective cover preserving direct summands, if and only if every (cyclic) R-module having a projective cover preserving direct summands is lifting. It is also proved that every cyclic module over a ring R is ⊕-supplemented if and only if every cyclic R-module is a direct sum of local modules. Consequently, a ring R is artinian serial if and only if every left and right R-module is a direct sum of local modules.     

On commutative rings which are strongly prüfer

Abstract

As defined by Nicholson [Nicholson, W. K. (1977). Lifting idempotents and exchange rings. Trans. Amer. Math. Soc. 229:269–278] an element of a ring R is clean if it is the sum of a unit and an idempotent, and a subset A of R is clean if every element of A is clean. It is shown that a semiprimitive Gelfand ring R is clean if and only if Max(R) is zero-dimensional; if and only if for each M ∈ Max(R), the intersection all prime ideals contained in M is generated by a set of idempotents. We also give several equivalent conditions for clean functional rings. In fact, a functional ring R is clean if and only if the set of clean elements is closed under sum; if and only if every zero-divisor is clean; if and only if; R has a clean prime ideal.     

On FC-Purity and I-Purity of Modules and Köthe Rings

In this article, several characterizations of certain classes of rings via FC-purity and I-purity are considered. Among others results, it is shown that every I-pure injective left R-module is projective if and only if every FC-pure projective left R-module is injective, if and only if, R is a semisimple ring. In particular, the structures of FC-pure projective and I-pure projective modules over a left Artinian ring are completely described. Also, it is shown that every left R-module is FC-pure projective if and only if every indecomposable left R-module is a finitely presented cyclic R-module, if and only if, R is a left Köthe ring. Finally, we introduce FC-pure flatness and I-pure flatness of modules and several characterizations of these notions are given. In particular, we show that a commutative ring R is quasi-Frobenius if and only if R is an Artinian ring and I-pure injective, if and only if, R is an Artinian ring and the injective envelope E(R) is an FC-pure projective R-module.     

Flat modules over valuation rings

Let R be a valuation ring and let Q be its total quotient ring. It is proved that any singly projective (respectively flat) module is finitely projective if and only if Q is maximal (respectively artinian). It is shown that each singly projective module is a content module if and only if any non-unit of R is a zero-divisor and that each singly projective module is locally projective if and only if R is self-injective. Moreover, R is maximal if and only if each singly projective module is separable, if and only if any flat content module is locally projective. Necessary and sufficient conditions are given for a valuation ring with non-zero zero-divisors to be strongly coherent or π-coherent.A complete characterization of semihereditary commutative rings which are π-coherent is given. When R is a commutative ring with a self-FP-injective quotient ring Q, it is proved that each flat R-module is finitely projective if and only if Q is perfect.     

Weakly regular rings with ACC on annihilators and maximality of strongly prime ideals of weakly regular rings

Ramamurthi proved that weak regularity is equivalent to regularity and biregularity for left Artinian rings. We observe this result under a generalized condition. For a ring R satisfying the ACC on right annihilators, we actually prove that if R is left weakly regular then R is biregular, and that R is left weakly regular if and only if R is a direct sum of a finite number of simple rings. Next we study maximality of strongly prime ideals, showing that a reduced ring R is weakly regular if and only if R is left weakly regular if and only if R is left weakly π-regular if and only if every strongly prime ideal of R is maximal.     

Mal'cev categories and fibration of pointed objects

The fibration p of pointed objects of a category E is shown to have some classifying properties: it is additive if and only if E is naturally Mal'cev, it is unital if and only if E is Mal'cev. The category E is protomodular if and only if the change of base functors relative to p are conservative.     

Weakly automorphism invariant modules and essential tightness

While a module is pseudo-injective if and only if it is automorphism-invariant, it was not known whether automorphism-invariant modules are tight. It is shown that weakly automorphism-invariant modules are precisely essentially tight. We give various examples of weakly automorphism-invariant and essentially tight modules and study their properties. Some particular results: (1) R is a semiprime right and left Goldie ring if and only if every right (left) ideal is weakly injective if and only if every right (left) ideal is weakly automorphism invariant; (2) R is a CEP-ring if and only if R is right artinian and every indecomposable projective right R-module is uniform and essentially R-tight.     

The Simple Connectedness of a Tame Weakly Shod Algebra

Abstract

We prove that a tame weakly shod algebra A which is not quasi-tilted is simply connected if and only if the orbit graph of its pip-bounded component is a tree, or if and only if its first Hochschild cohomology group H¹(A) with coefficients in _A A _A vanishes. We also show that it is strongly simply connected if and only if the orbit graph of each of its directed components is a tree, or if and only if H¹(A) = 0 and it contains no full convex subcategory which is hereditary of type 𝔸?, or if and only if it is separated and contains no full convex subcategory which is hereditary of type 𝔸?.     

Hölder Conditions for Endomorphisms of Hyperbolic Groups

It is proved that an endomorphism φ of a hyperbolic group G satisfies a Hölder condition with respect to a visual metric if and only if φ is virtually injective and Gφ is a quasiconvex subgroup of G. If G is virtually free or torsion-free co-hopfian, then φ is uniformly continuous if and only if it satisfies a Hölder condition if and only if it is virtually injective.     

Gorenstein injective and strongly cotorsion modules

By investigating the properties of some special covers and envelopes of modules, we prove that if R is a Gorenstein ring with the injective envelope of _R R flat, then a left R-module is Gorenstein injective if and only if it is strongly cotorsion, and a right R-module is Gorenstein flat if and only if it is strongly torsionfree. As a consequence, we get that for an Auslander-Gorenstein ring R, a left R-module is Gorenstein injective (resp. flat) if and only if it is strongly cotorsion (resp. torsionfree).     

Finite groups with trivial Frattini subgroup

All groups considered are finite. A group has a trivial Frattini subgroup if and only if every nontrivial normal subgroup has a proper supplement.The property is normal subgroup closed, but neither subgroup nor quotient closed. It is subgroup closed if and only if the group is elementary, i.e. all Sylow subgroups are elementary abelian. If G is solvable, then G and all its quotients have trivial Frattini subgroup if and only if every normal subgroup of G has a complement. For a nilpotent group, every nontrivial normal subgroup has a supplement if and only if the group is elementary abelian. Consequently, the center of a group in which every normal subgroup has a supplement is an elementary abelian direct factor.     

Modules over Endomorphism Rings

It is proved that A is a right distributive ring if and only if all quasiinjective right A-modules are Bezout left modules over their endomorphism rings if and only if for any quasiinjective right A-module M which is a Bezout left End (M)-module, every direct summand N of M is a Bezout left End(N)-module. If A is a right or left perfect ring, then all right A-modules are Bezout left modules over their endomorphism rings if and only if all right A-modules are distributive left modules over their endomorphism rings if and only if A is a distributive ring.     

Trivial Ring Extensions Defined by Arithmetical-Like Properties

In this article we investigate the transfer of the notions of elementary divisor ring, Hermite ring, Bezout ring, and arithmetical ring to trivial ring extensions of commutative rings by modules. Namely, we prove that the trivial ring extension R: = A ? B defined by extension of integral domains is an elementary divisor ring if and only if A is an elementary divisor ring and B = qf(A); and R is an Hermite ring if and only if R is a Bezout ring if and only if A is a Bezout domain and qf(A) = B. We provide necessary and sufficient conditions for R = A ? E to be an arithmetical ring when E is a nontorsion or a finitely generated A ? module. As an immediate consequences, we show that A ? A is an arithmetical ring if and only if A is a von Neumann regular ring, and A ? Q(A) is an arithmetical ring if and only if A is a semihereditary ring.     

Locality Criteria for Cocommutative Hopf Algebras

We prove that a finite-dimensional cocommutative Hopf algebra H is local, if and only if the subalgebra generated by the first term of its coradical filtration H ₁ is local. In particular if H is connected, H is local if and only if all the primitive elements of H are nilpotent.     

Bézoutians and injectivity of polynomial maps

We prove that an endomorphism f of affine space is injective on rational points if its Bézoutian is constant. Similarly, f is injective at a given rational point if its reduced Bézoutian is constant. We also show that if the Jacobian determinant of f is invertible, then f is injective at a given rational point if and only if its reduced Bézoutian is constant.     

Ends of graphed equivalence relations,I

Given a countable Borel equivalence relation E on a Polish space, we show: (1) E admits an endless graphing if and only if E is smooth; (2) E admits a locally finite single-ended graphing if and only if E is aperiodic; (3) E admits a graphing for which there is a Borel way of selecting two ends from each component if and only if E is hyperfinite; and (4) E admits a graphing for which there is a Borel way of selecting a finite set of at least three ends from each component if and only if E is smooth. The author was supported in part by NSF VIGRE Grant DMS-0502315.     

Summability and vector amarts

We show that an operator is absolutely summing if and only if it maps amarts into uniform amarts, from which we can deduce a theorem of A. Bellow and another of Edgar-Sucheston. We also show that the absolute value of a Banach lattice valued potential is a potential if and only if the lattice is an A-M space from which we deduce that the L¹-bounded amarts form a Ricsz space if and only if the space is finite dimensional.     

When is c(x) a Clean Ring?

An element of a ring R is called clean if it is the sum of a unit and an idempotent and a subset A of R is called clean if every element of A is clean. A topological characterization of clean elements of C(X) is given and it is shown that C(X) is clean if and only if X is strongly zero-dimensional, if and only if there exists a clean prime ideal in C(X). We will also characterize topological spaces X for which the ideal C_K(X) is clean. Whenever X is locally compact, it is shown that C_K(X) is clean if and only if X is zero-dimensional.     

Minimal harmoniously colorable designs

A graph G is minimal harmoniously colorable if it has a proper vertex coloring in which each pair of colors occurs exactly once on an edge. In particular, if D is a 2-design we consider the graph G whose vertices are the points and blocks of D and where two vertices of G are adjacent if and only if the corresponding elements of D are incident. It will be shown that if D is symmetric then G is minimal harmoniously colorable if and only if D is a Hadamard design with corresponding Hadamard matrix of a certain form. We obtain some results if D is nonsymmetric, and construct two classes of nonsymmetric minimal harmoniously colorable designs. © 1994 John Wiley & Sons, Inc.